The fundamental theorem of algebra

Every polynomial equation of degree n with complex coefficients has n roots in the complex numbers.

In fact there are many equivalent formulations: for example that every real polynomial can be expressed as the product of real linear and real quadratic factors.

Early studies of equations by al-Khwarizmi (c 800) only allowed positive real roots and the FTA was not relevant. Cardan was the first to realise that one could work with quantities more general than the real numbers. This discovery was made in the course of studying a formula which gave the roots of a cubic equation. The formula when applied to the equation x3 = 15x + 4 gave an answer involving √-121 yet Cardan knew that the equation had x = 4 as a solution. He was able to manipulate with his 'complex numbers' to obtain the right answer yet he in no way understood his own mathematics.

Bombelli, in his Algebra, published in 1572, was to produce a proper set of rules for manipulating these 'complex numbers'. Descartes in 1637 says that one can 'imagine' for every equation of degree n, n roots but these imagined roots do not correspond to any real quantity.

Viète gave equations of degree n with n roots but the first claim that there are always n solutions was made by a Flemish mathematician Albert Girard in 1629 in L'invention en algèbre . However he does not assert that solutions are of the form a + bi, a, b real, so allows the possibility that solutions come from a larger number field than C. In fact this was to become the whole problem of the FTA for many years since mathematicians accepted Albert Girard's assertion as self-evident. They believed that a polynomial equation of degree n must have n roots, the problem was, they believed, to show that these roots were of the form a + bi, a, b real.

Now Harriot knew that a polynomial which vanishes at t has a root x - t but this did not become well known until stated by Descartes in 1637 in La géométrie, so Albert Girard did not have much of the background to understand the problem properly.

A 'proof' that the FTA was false was given by Leibniz in 1702 when he asserted that x4 + t4 could never be written as a product of two real quadratic factors. His mistake came in not realising that √i could be written in the form a + bi, a, b real.

D'Alembert in 1746 made the first serious attempt at a proof of the FTA. For a polynomial f he takes a real b, c so that f(b) = c. Now he shows that there are complex numbers z1 and w1 so that

|z1| < |c|, |w1| < |c|.

He then iterates the process to converge on a zero of f. His proof has several weaknesses. Firstly, he uses a lemma without proof which was proved in 1851 by Puiseau, but whose proof uses the FTA! Secondly, he did not have the necessary knowledge to use a compactness argument to give the final convergence. Despite this, the ideas in this proof are important.

Euler was soon able to prove that every real polynomial of degree n, n ≤ 6 had exactly n complex roots. In 1749 he attempted a proof of the general case, so he tried to proof the FTA for Real Polynomials:

Every polynomial of the nth degree with real coefficients has precisely n zeros inC.

His proof in Recherches sur les racines imaginaires des équations is based on decomposing a monic polynomial of degree 2n into the product of two monic polynomials of degree m = 2n-1. Then since an arbitrary polynomial can be converted to a monic polynomial by multiplying by axk for some k the theorem would follow by iterating the decomposition. Now Euler knew a fact which went back to Cardan in Ars Magna, or earlier, that a transformation could be applied to remove the second largest degree term of a polynomial. Hence he assumed that

and then multiplied up and compared coefficients. This Euler claimed led to g, h, ... being rational functions of A, B, ..., t. All this was carried out in detail for n = 4, but the general case is only a sketch.

In 1772 Lagrange raised objections to Euler's proof. He objected that Euler's rational functions could lead to 0/0. Lagrange used his knowledge of permutations of roots to fill all the gaps in Euler's proof except that he was still assuming that the polynomial equation of degree n must have n roots of some kind so he could work with them and deduce properties, like eventually that they had the form a + bi, a, b real.

Laplace, in 1795, tried to prove the FTA using a completely different approach using the discriminant of a polynomial. His proof was very elegant and its only 'problem' was that again the existence of roots was assumed.

Gauss is usually credited with the first proof of the FTA. In his doctoral thesis of 1799 he presented his first proof and also his objections to the other proofs. He is undoubtedly the first to spot the fundamental flaw in the earlier proofs, to which we have referred many times above, namely the fact that they were assuming the existence of roots and then trying to deduce properties of them. Of Euler's proof Gauss says

... if one carries out operations with these impossible roots, as though they really existed, and says for example, the sum of all roots of the equationxm+axm-1 + bxm-2 + . . . = 0 is equal to -a even though some of them may be impossible (which really means: even if some are non-existent and therefore missing), then I can only say that I thoroughly disapprove of this type of argument.

Gauss himself does not claim to give the first proper proof. He merely calls his proof new but says, for example of d'Alembert's proof, that despite his objections

a rigorous proof could be constructed on the same basis.

Gauss's proof of 1799 is topological in nature and has some rather serious gaps. It does not meet our present day standards required for a rigorous proof.

In 1814 the Swiss accountant Jean Robert Argand published a proof of the FTA which may be the simplest of all the proofs. His proof is based on d'Alembert's 1746 idea. Argand had already sketched the idea in a paper published two years earlier Essai sur une manière de représenter les quantitiés imaginaires dans les constructions géometriques. In this paper he interpreted i as a rotation of the plane through 90° so giving rise to the Argand plane or Argand diagram as a geometrical representation of complex numbers. Now in the later paper Réflexions sur la nouvelle théorie d'analyseArgand simplifies d'Alembert's idea using a general theorem on the existence of a minimum of a continuous function.

In 1820 Cauchy was to devote a whole chapter of Cours d'analyse to Argand's proof (although it will come as no surprise to anyone who has studied Cauchy's work to learn that he fails to mention Argand !) This proof only fails to be rigorous because the general concept of a lower bound had not been developed at that time. The Argand proof was to attain fame when it was given by Chrystal in his Algebra textbook in 1886. Chrystal's book was very influential.

Two years after Argand's proof appeared Gauss published in 1816 a second proof of the FTA. Gauss uses Euler's approach but instead of operating with roots which may not exist, Gauss operates with indeterminates. This proof is complete and correct.

A third proof by Gauss also in 1816 is, like the first, topological in nature. Gauss introduced in 1831 the term 'complex number'. The term 'conjugate' had been introduced by Cauchy in 1821.

Gauss's criticisms of the Lagrange-Laplace proofs did not seem to find immediate favour in France. Lagrange's 1808 2nd Edition of his treatise on equations makes no mention of Gauss's new proof or criticisms. Even the 1828 Edition, edited by Poinsot, still expresses complete satisfaction with the Lagrange-Laplace proofs and no mention of the Gauss criticisms.

In 1849 (on the 50th anniversary of his first proof!) Gauss produced the first proof that a polynomial equation of degree n with complex coefficients has n complex roots. The proof is similar to the first proof given by Gauss. However it is adds little since it is straightforward to deduce the result for complex coefficients from the result about polynomials with real coefficients.

It is worth noting that despite Gauss's insistence that one could not assume the existence of roots which were then to be proved reals he did believe, as did everyone at that time, that there existed a whole hierarchy of imaginary quantities of which complex numbers were the simplest. Gauss called them a shadow of shadows.

It was in searching for such generalisations of the complex numbers that Hamilton discovered the quaternions around 1843, but of course the quaternions are not a commutative system. The first proof that the only commutative algebraic field containing R was given by Weierstrass in his lectures of 1863. It was published in Hankel's book Theorie der complexen Zahlensysteme.

Of course the proofs described above all become valid once one has the modern result that there is a splitting field for every polynomial. Frobenius, at the celebrations in Basle for the bicentenary of Euler's birth said:-

Euler gave the most algebraic of the proofs of the existence of the roots of an equation, the one which is based on the proposition that every real equation of odd degree has a real root. I regard it as unjust to ascribe this proof exclusively to Gauss, who merely added the finishing touches.

The Argand proof is only an existence proof and it does not in any way allow the roots to be constructed. Weierstrass noted in 1859 made a start towards a constructive proof but it was not until 1940 that a constructive variant of the Argand proof was given by Hellmuth Kneser. This proof was further simplified in 1981 by Martin Kneser, Hellmuth Kneser's son.